Introduction to Thermodynamics PDF

Introduction to thermodynamics Degree Industrial Technologies Engineering Lesson 7: Introduction to Thermodynamics 7.1 Thermodynamics: concepts and definitions 7.2 Equilibrium states: Quasi-static processes and reversible processes 7.3 Work 7.4 Temperature 7....

Introduction to thermodynamics Degree Industrial Technologies Engineering Lesson 7: Introduction to Thermodynamics 7.1 Thermodynamics: concepts and definitions 7.2 Equilibrium states: Quasi-static processes and reversible processes 7.3 Work 7.4 Temperature 7.5 Thermometry. Ideal gas scale 7.6 Thermal coefficients Introduction to thermodynamics Degree Industrial Technologies Engineering 7.1 Thermodynamics: concepts and definitions Thermo: Heat The Word thermodynamics comes from greek Dynamics: Power In principle thermodynamics would be the part of physics devoted to HEAT Or thermodynamics is devoted to the transformation of heat into work HEAT WORK In fact, the energy exists in many different forms: light energy, heat energy, mechanical energy, gravitational energy, nuclear energy, …. In general, the goal of thermodynamics would be One form of energy Another form of energy Introduction to thermodynamics Degree Industrial Technologies Engineering Thermodynamics laws  Zeroth law of thermodynamics: it allows us to define the concept of temperature  First law of thermodynamics: it is about the conservation of energy  Second law of thermodynamics: It allows us to define the concept of entropy or the direction in which processes can take place  Third law of thermodynamics: It allows us to interpret entropy as the order of the systems Introduction to thermodynamics Degree Industrial Technologies Engineering System It is the portion of the Universe that will be studied. It is separated from the surroundings by a boundary The surroundings is the rest of the Universe Boundary The boundary: The system interacts with the surroundings through the boundary: System exchanging heat, work, matter Surroundings The system can be described in two different ways: Microscopic description: The system is formed by atoms or molecules. We try to describe the evolution of each atom or molecule of the system: position, velocity, acceleration,… It is impossible to carry out this task. So, in this case a statistical description is done: Statistical thermodynamics. Macroscopic description: The system is described by using certain macroscopic physical quantities: Temperature, volume, density,… In this physics course a macroscopic description will be considered Introduction to thermodynamics Degree Industrial Technologies Engineering Systems can be classified in different ways: a) On considering the type of boundary ISOLATED SYSTEM: It is one to which it cannot be added neither energy (heat/work) nor matter. Example: Thermos CLOSED SYSTEM: It is one to which it can be added energy (heat/work) but no matter. Example: A closed bottle OPEN SYSTEM: It is one to which it can be added energy (heat/work) and matter. Example: An open bottle Introduction to thermodynamics Degree Industrial Technologies Engineering b) On considering what is inside the system HOMOGENEOUS SYSTEM: There is a uniform or single phase in the system, such as a solid, a liquid or a gas. The phase can be a mixture of two or more substances, but the composition does not vary from one point to another one. INHOMOGENEOUS SYSTEM: There is more than one phase, such as a solid plus liquid, liquid plus gas Introduction to thermodynamics Degree Industrial Technologies Engineering Physical quantities In physics, the physical properties are associated with physical magnitudes (Physical quantities) that can be measured and expressed with a number. The physical quantities are classified:  Intensive physical quantity: It is associated with a bulk property. It does not depend on the size of the system nor on the amount of matter of the system. Examples: Pressure, Temperature, Density  Extensive physical quantity: It depends on the size and amount of mass of the system. It is an additive quantity. So, if the system is divided into several subsystems, the value of the physical quantity in each system is different, and the value of the physical quantity in the system is the sum of the values that physical quantity acquires in each subsystem. Examples: Energy, Entropy, Mass. Notes: An extensive physical quantity can be transformed into an intensive physical quantity by dividing by the volume: Introduction to thermodynamics Degree Industrial Technologies Engineering Note: An extensive physical quantity can be transformed into an intensive physical quantity by dividing by the volume: Extensive quantity Intensive quantity Mass (M) Density: r=M/V Heat Capacity: CP Specific heat capacity: cP=CP/V Amount of a substance: mol Molality: b = mol/mass Molar concentration: c=Mol/V Introduction to thermodynamics Degree Industrial Technologies Engineering We will learn to distinguish among: state variable, state function and state equation In thermodynamics a system is under study. The system is characterized by Thermodynamic properties. There is a set of parameters which are used to characterize the thermodynamic properties. Those parameters which can be experimentally measured are the STATE VARIABLES Examples of state variables: Pressure (P), Temperature (T), Volume (V), amount of moles (n) Not all the state variables are used to characterize the system, only a few of them which are independent. A STATE FUNCTION is also a parameter that characterizes the thermodynamic properties of the system, but we cannot measure it experimentally. However, it can be written in terms of a certain number of state variables. Examples of state functions: Internal energy (U), Enthalpy (H), Gibbs function (G), Free Energy (F). For an ideal gas U=U(T) A STATE EQUATION is a mathematical equation written in terms of a certain number of state variables. This equation characterizes the state of the system under study. The equation involves those state equations which are enough for characterizing the state of the system Example: The state equation for an ideal gas: PV=nRT Introduction to thermodynamics Degree Industrial Technologies Engineering Note: Heat and work are not state variables nor state functions: Introduction to thermodynamics Degree Industrial Technologies Engineering Ideal Gas Ideal gas: It is a system formed by particles (molecules) that do not interact between them. When the particles collide between them or with the walls , the collisions are elastic P: Pressure 𝑃𝑉𝑚 T: Temperature It is verified: lim = 𝑅 (1) Vm: volume occupied by one mole 𝑃→0 𝑇 R: Gas constant = 8,31451 J/(mole·K) In fact (1) is the Avogadro’s law: All of the gases at the same pressure, temperature and volume have the same number of moles. 1 mole: There are so many atoms or molecules as in 12 g of 12C; that is to say: NA = 6,022·1023 atoms. NA is the Avogadro’s number 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑚𝑎𝑠𝑠 (𝑔𝑟𝑎𝑚𝑠) 𝑛 = 𝑚𝑜𝑙𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 = = 𝑁𝐴 1 𝑚𝑜𝑙𝑒 𝑚𝑎𝑠𝑠 Introduction to thermodynamics Degree Industrial Technologies Engineering P: Pressure in N/m2 (Pascal) T: Temperature in Kelvin (K) State equation for an ideal gas 𝑃𝑉 = 𝑁𝐾𝐵 𝑇 V: volume in m3 N: number of molecules KB: Boltzman constant; KB=1,38·10-23 J/K 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 It is verified: 𝑁 = 𝑚𝑜𝑙𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛 𝑥 𝑁𝐴 = 𝑛𝑁𝐴 𝑚𝑜𝑙𝑒 𝑃𝑉 = 𝑁𝐾𝐵 𝑇 = 𝑛𝑁𝐴 𝐾𝐵 𝑇 = 𝑛𝑅𝑇 With R = NAKB It is usually used the following equation: 𝑃𝑉 = 𝑛𝑅𝑇 P: in atmospheres T: Temperature in Kelvin (K) V: in litres n: number of moles R: Gases constant; R=0,08206 (atm·l)/(mole· K) In normal conditions (P = 1 atm; T = 273,15 K), there are only a few gases that can be considered ideal gases. However, at low pressure (P approaches zero) all of the gases exhibit an ideal behavior Introduction to thermodynamics Degree Industrial Technologies Engineering 7.2 Equilibrium states: quasi-static and reversible processes A system is in thermodynamic equilibrium if there is no tendency to a spontaneous change. There are different types of equilibrium:  Mechanic equilibrium: The sum of all the forces acting on each particle of the system is zero (External forces plus internal forces)  Thermal equilibrium: The temperature has the same value everywhere in the system it does not change. In case the system is in thermal contact with the surroundings, it has the same temperature as the surroundings.  Phases equilibrium: the mass of each phase present in the system has achieved a certain value and it does not change  Chemical equilibrium: The chemical composition does not change (no chemical reactions) A system is in thermodynamic equilibrium if: it is in mechanic equilibrium, thermal equilibrium, phases equilibrium, chemical equilibrium,… Introduction to thermodynamics Degree Industrial Technologies Engineering Process: When a system undergoes a change from one thermodynamic state to another one, because some state variables have changed such as volume, temperature, pressure, then it is said that the system has undergone a thermodynamic process. On changing the system from one state, denoted as 1, to another state, denoted as 2, it passes through different states which define a path or trajectory. Type of Processes when the system goes from state 1 to 2: Quasi-static: All of the intermediate states of the path are equilibrium states. The process takes a very long time, so that the system achieves thermodynamic equilibrium in all the states of the path. Non-static: The intermediate states of the path are not equilibrium states. Reversible: It is a quasi-static process so that the system can go from state 2 to 1 passing through the same intermediate states. Irreversible: The system does the inverse process but the initial state is not achieved Introduction to thermodynamics Degree Industrial Technologies Engineering In case some variable remains constant, the processes are defined Isobaric process: The pressure remains constant Isochoric process: The volume remains constant Isothermal process: The temperature remains constant Adiabatic process: No heat is added or removed from the system during the process These processes are only possible if the boundary exhibits certain features: What is allowed? Yes No The volume Mobile boundary Fixed boundary changes Exchanges of heat Diathermal Adiabatic boundary boundary Exchanges of mass Permeable Impermeable boundary boundary Introduction to thermodynamics Degree Industrial Technologies Engineering Processes representations. P-V diagrams Processes are represented by diagrams. In this way it is easier to realize how the process has been carried out. Examples: P-V diagrams (Pressure-Volume) P-T diagrams (Pressure-Temperature) Examples for an ideal gas: 𝑃𝑉 = 𝑛𝑅𝑇 P P 1 2 Find out a relationship 1 P1 P1 between Ti and Tf P3 3 4 P4 V V V1 V2 V1 V4 Process 1 → 2 Isobaric: P is constant Process 1→4 Isothermal: T is constant Process 1 → 3 Isochoric: V is constant Introduction to thermodynamics Degree Industrial Technologies Engineering 7.3 Work PRESSURE Pressure is defined as the force per unit of surface. However only the component of the force perpendicular to the surface is considered Units of pressure 𝐹 𝑁𝑒𝑤𝑡𝑜𝑛 𝑃= 𝐹⊥ 𝐹 SI: P𝑎𝑠𝑐𝑎𝑙 = 𝑆 𝑃 = 𝑚2 F S F S 𝑆 𝑃 = 𝑆 𝑑𝑦𝑛𝑒 F⊥ CGS: 𝐵𝑎𝑟𝑦𝑒 = 𝑐𝑚2 1 Barye is very small, so it is defined a new unit: 1 bar. Then: 1 bar = 1·106 Barye Another unit of pressure is the ATMOSPHERE. It is defined as the pressure exerted by a 760 mm column of Hg on a surface of 1 mm2 at 0 oC. Unit 1 Unit 2 1 bar 1·106 Barye 1 bar 1·105 Pascal 1 atmosphere (atm) 101325 Pascal 1 atmosphere (atm) 1,01325 bar Introduction to thermodynamics Degree Industrial Technologies Engineering On speaking about pressure: Absolute pressure: It is the pressure measured with respect to vacuum (PV=0) Gauge pressure or manometric pressure: It is the pressure measured with respect to the atmosphere pressure Vacuum pressure: It is the pressure below the atmospheric pressure Absolute pressure P manometric Atmospheric pressure Atmospheric pressure P vacuum P abs Absolute pressure P atm P abs P atm Absolute vacuum Absolute vacuum 𝑃𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 = 𝑃𝑎𝑏𝑠 − 𝑃𝑎𝑡𝑚 𝑃𝑣𝑎𝑐𝑢𝑢𝑚 = 𝑃𝑎𝑡𝑚 − 𝑃𝑎𝑏𝑠 Introduction to thermodynamics Degree Industrial Technologies Engineering Manometer P manometric 𝑃𝑎𝑏𝑠 = 𝑃𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 + 𝑃𝑎𝑡𝑚 P abs Introduction to thermodynamics Degree Industrial Technologies Engineering Pressure in a liquid The pressure on a liquid varies with depth. Pat The point O must be in equilibrium, so the net force acting on O must be zero: 𝑃𝑆 − 𝑃𝑎𝑡 𝑆 − 𝜌𝑔𝑧𝑆 = 0 ⟹ 𝑃 = 𝑃𝑎𝑡 +𝜌𝑔𝑧 z O P Force exerted by Force exerted by the Weight of the S is the surface of the liquid which is atmospheric pressure column of liquid the column below O Pascal’s law: a change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container Pressure due to a gas  The pressure on the walls of the container is due to the collisions of the molecules of the gas on the walls  If there are different gases in the container, each gas exerts a partial pressure. The total pressure is the sum of the partial pressures. Introduction to thermodynamics Degree Industrial Technologies Engineering Torricelli experiment: how can the atmospheric pressure be measured ? 𝑃𝐶 = 0 The pressure at B is: C 𝑃𝐵 𝑆 −𝑃𝐶 𝑆 − 𝜌𝑔𝑧𝑆 = 0 ⟹ 𝑃𝐵 = 𝜌𝑔𝑧 The pressure at D is the atmospheric pressure. Z=760 mm Hg As PB=PD 𝑃𝐵 = 𝑃𝐷 = 𝜌𝑔𝑧 D B S is the surface of r(Hg)=13,6 g/cm3 the column Introduction to thermodynamics Degree Industrial Technologies Engineering Process at constant pressure Let us consider a gas inside a container. The container is closed by a piston that can move. There is weight on the piston The piston must be in equilibrium, so the sum of all the forces acting on the piston must be zero. Block in equilibrium: Fatm N 𝑁 = 𝑊𝐵 Piston in equilibrium: N Fgas WB 𝐹𝑎𝑡𝑚 +𝑁 + 𝑊𝑃 −𝐹𝑔𝑎𝑠 = 0 WP The force exerted by the gas on the piston is: 𝐹𝑔𝑎𝑠 = 𝑊𝑃 + 𝑊𝐵 + 𝐹𝑎𝑡𝑚 𝐹 𝑊 𝑊 𝐹 𝑊 𝑊 The pressure of the gas: 𝑃𝐺𝑎𝑠 = 𝐺𝑎𝑠 = 𝑃 + 𝐵 + 𝑎𝑡𝑚 = 𝑃 + 𝐵 + 𝑃𝑎𝑡𝑚 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 What does it happen if the temperature changes? The pressure does not change 𝑛𝑅𝑇 As 𝑃𝑉 = 𝑛𝑅𝑇 ⟹ 𝑃 = = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 The volume must change 𝑉 If T increases, the volume must increases and the gas is expanded If T decreases, the volume must decrease and the gas undergoes a compression Introduction to thermodynamics Degree Industrial Technologies Engineering Work concept In mechanics the work done on a particle is defined as: d means that in general the work is 𝛿𝑊 = 𝐹 · 𝑑𝑟 irreversible: it depends on the path In thermodynamics, the work is related to the work done by the external forces (associated with the surroundings) acting on the system Sign criteria : If the system “wins” energy the work is positive and if the system loses energy the work is negative dW > 0 The work is done by the surroundings on the system dW < 0 The work is done by the system on the surroundings  Introduction to thermodynamics Degree Industrial Technologies Engineering Let us consider a cylinder with a piston. There is a gas inside. The system goes from state 1 to state 2 and all of the intermediate states are equilibrium states FGas Fext FGas Fext x x dx P,V,T,n There is equilibrium at any state: 𝐹𝐺𝑎𝑠𝑖 − 𝐹𝑒𝑥𝑡𝑖 = 0 ⟹ 𝐹𝐺𝑎𝑠 = 𝐹𝑒𝑥𝑡 𝛿𝑊 = 𝐹 𝑒𝑥𝑡 · 𝑑𝑟 = −𝐹𝑒𝑥𝑡 𝑖 · 𝑑𝑥𝑖 = −𝐹𝑒𝑥𝑡 𝑑𝑥 On the other hand: 𝐹𝐺𝑎𝑠 = 𝑃𝑆 P is the pressure of the gas and S is the piston surface 𝐹𝑒𝑥𝑡 = 𝐹𝐺𝑎𝑠 = 𝑃𝑆 𝛿𝑊 = −𝑃𝑆𝑑𝑥 = −𝑃𝑑𝑉 Therefore: 𝛿𝑊 = −𝑃𝑑𝑉 ⟹ 𝑊 = − 𝑃𝑑𝑉  Introduction to thermodynamics Degree Industrial Technologies Engineering Some examples (Ideal gas) Isobaric process P=Constant Isochoric process P P V= Constant 2 𝑉2 1 2 P1 1 𝑊=− 𝑃𝑑𝑉 = −𝑃 𝑑𝑉 = P1 1 𝑉1 𝛿𝑊 = −𝑃𝑑𝑉 = 0 = −𝑃(𝑉2 − 𝑉1) P2 2 As dV=0 V V V1 V2 V1 Isothermal process 2 P T = Constant 𝑛𝑅𝑇 𝑊=− 𝑃𝑑𝑉 For an ideal gas 𝑃𝑉 = 𝑛𝑅𝑇 ⟹ 𝑃 = 1 1 𝑉 P1 Therefore: 2 2 𝑛𝑅𝑇 𝑑𝑉 𝑉2 𝑊=− 𝑑𝑉 = −𝑛𝑅𝑇 = −𝑛𝑅𝑇𝐿𝑛 = 1 𝑉 1 𝑉 𝑉1 2 𝑉1 P2 = 𝑛𝑅𝑇𝐿𝑛 𝑉2 V V1 V2 Introduction to thermodynamics Degree Industrial Technologies Engineering 7.4 Temperature Some ideas about temperature:  The temperature is related to the internal energy of a system. For a gas: The temperature is related to the kinetic energy of the particles of the system For a solid: The temperature is related to the motion of the atoms around their equilibrium positions  The temperature is an intensive variable: In thermal equilibrium all the points of the system have the same temperature. It does not depend on the amount of mass of the system Zeroth law of thermodynamics: 1) Two isolated systems which are in 2) If systems A and B are in thermal contact through a diathermic wall, after a equilibrium with a third system C, them A time, that can be as long as it is necessary, and B are also in thermal equilibrium both systems are in thermal equilibrium Adiabatic A B wall Adiabatic wall A B C Diathermic Diathermic wall wall 𝑇𝐴 = 𝑇𝐵 𝑇𝐴 = 𝑇𝐶 = 𝑇𝐵 Introduction to thermodynamics Degree Industrial Technologies Engineering Note: Our finger is not a good thermometer or a good device for measuring the temperature Only feeling: it is hot or it is cold. It is only useful for preventing to get burned Let us consider a classroom which is in thermal equilibrium: all of the objects inside have the same temperature Important: -If there is a temperature difference there is a transference of thermal energy -The transference of the thermal energy depends on the thermal conductivity. Conclusion Wrong!!!!! Tmetal< Twood On touching the wood : it is hot On touching the metal: it is cold Tmetal= Twood The thermal conductivity of the metal is greater than that of the wood Introduction to thermodynamics Degree Industrial Technologies Engineering 7.5 Thermometry. Ideal gas scale Thermometer: It is a device that allows us to measure the temperature of a body The thermometer is based in a property of a certain substance. The physical property is named thermometric property. It usually exhibits a linear dependence with the temperature Examples  For an ideal gas. Physical property: Volume (the pressure remains constant); Pressure (The volume remains constant)  For a liquid. Physical property: Volume (Hg, mercury)  For a metallic rod. Physical property: Electrical resistance Most important features of the thermometric property  The physical property only depends on temperature  There is a biunivocal relation between temperature and the physical property  The thermal equilibrium between the thermometer and the body is rapidly achieved  The measurements are reproducible Introduction to thermodynamics Degree Industrial Technologies Engineering Let us build a thermometer!!!! X is the thermometric property. It has a linear dependence with respect to the temperature. 𝑡 = 𝑎𝑥 + 𝑏 a and b are constants Temperature Thermometric property Calibration Water melting point: 0 oC (Celsius scale) Water boiling point: 100 oC 0 = 𝑎𝑥𝑚 + 𝑏 100 100𝑥𝑚 𝑎= 𝑏=− 𝑥𝑏 − 𝑥𝑚 𝑥𝑏 − 𝑥𝑚 100 = 𝑎𝑥𝑏 + 𝑏 100 100𝑥𝑚 100 Therefore: 𝑡= 𝑥− = 𝑥 − 𝑥𝑚 𝑥𝑏 − 𝑥𝑚 𝑥𝑏 − 𝑥𝑚 𝑥𝑏 − 𝑥𝑚 Introduction to thermodynamics Degree Industrial Technologies Engineering Besides the Celsius scale there is other scales such as Fahrenheit scale Mixture :ice + Boiling water Scale liquid water 1 atmosphere divisions Celsius 0 oC 100 oC 100 Fahrenheit 32 oF 212 oF 180 180 o 100 o 100 o𝐶 = 180 o𝐹 ⇒ 1o𝐶 = 𝐹; 1 𝐹 = o 𝐶 100 180 So: 180 o Passing from Celsius degrees to 𝑇 o𝐹 = 32 + 𝑡 𝐶 Fahrenheit degrees 100 100 Passing from Fahrenheit degrees to 𝑡 o𝐶 = 𝑇 o𝐹 − 32 180 Celsius degrees Introduction to thermodynamics Degree Industrial Technologies Engineering Absolute temperature scale Let us consider a thermometer. It is an ideal gas at constant volume Pa 𝑃𝑉 𝑃𝑉 = 𝑛𝑅𝑇 ⇒ 𝑇 = 𝑛𝑅 h Hg Reservoir. It is used P A Hg to keep constant the Ideal gas volume occupied by the gas The value of the pressure of the gas will be: 𝑃 = 𝑃𝑎 + ℎ (𝑚𝑚 𝐻𝑔) Substance. The Once P is known, the temperature is 0 thermometer determined from measures its 𝑃𝑉 temperature 𝑇= 𝑛𝑅 Introduction to thermodynamics Degree Industrial Technologies Engineering Let it be the following experiment: There are two vessels, one with a mixture of liquid and ice water (0 oC) and other with a boiling water (100 o C). The temperature of the vessels is measured with the thermometer. First couple of measurements with n1 moles (1) 𝑛𝑖 𝑅 𝑃𝑖 = 𝑇𝑖 Second couple of measurements with n2 moles (2) 𝑉 Third couple of measurements with n3 moles (3) (3) P (2) (1) On prolonging the straight lines, they cut the abscissa (P=0)) at the same point: T = -273,15 o C -273,15 0 100 T(oC) Introduction to thermodynamics Degree Industrial Technologies Engineering At P = 0, the kinetic energy of the molecules is zero, so the molecules are at rest. A new scale of temperature is established, Kelvin scale. In this scale the, the zero is established at -273,15 degrees Celsius. In this scale, there are 100 divisions between the melting point and the boiling point of water 1 degree Kelvin = 1 degree Celsius 𝑇 o𝐾 = 𝑡 o𝐶 + 273.15 𝑡 o𝐶 = 𝑇 o𝐾 − 273.15 X General conference on weights and measures: It was established that the triple point of water (solid, liquid and gas) at 4,58 mm Hg in the Kelvin scale is 273.16 K. This point corresponds to 0,01 oC Introduction to thermodynamics Degree Industrial Technologies Engineering 7.7 Thermal coefficients Solid The volume of a solid varies on changing temperature. The atoms of the solid are vibrating around the equilibrium positions. The average distance between the atoms changes on changing the temperature. On increasing the temperature the average distances between the atoms also increase For a rod (solid with only one dimension) the linear thermal expansion coefficient is defined as: 1 𝜕𝐿 𝛼= 𝐿 𝜕𝑇 𝑃 Therefore: ∆𝐿 = 𝛼𝐿∆𝑇 If Lo is the initial length of the rod ∆𝐿 = 𝐿 − 𝐿𝑜 = 𝛼𝐿𝑜 ∆𝑇 ⇒ 𝐿 = 𝐿𝑜 + 𝛼𝐿𝑜 ∆𝑇 Introduction to thermodynamics Degree Industrial Technologies Engineering In three dimensions. The volume thermal expansion coefficient is defined as: 1 𝜕𝑉 𝛽= L3 𝑉 𝜕𝑇 𝑃 L2 L1 As 𝑉 = 𝐿1 𝐿2 𝐿3 𝜕𝑉 𝜕𝐿1 𝜕𝐿2 𝜕𝐿3 = 𝐿2 𝐿3 + 𝐿1 𝐿3 + 𝐿1 𝐿2 𝜕𝑇 𝑃 𝜕𝑇 𝑃 𝜕𝑇 𝑃 𝜕𝑇 𝑃 1 𝜕𝑉 1 𝜕𝐿1 1 𝜕𝐿2 1 𝜕𝐿3 𝛽= = + + 𝑉 𝜕𝑇 𝑃 𝐿1 𝜕𝑇 𝑃 𝐿2 𝜕𝑇 𝑃 𝐿3 𝜕𝑇 𝑃 a1 a2 a3 𝛽 = 𝛼 1 + 𝛼2 + 𝛼3 If the solid is isotropic, then: 𝛼1 = 𝛼2 = 𝛼3 𝛽 = 3𝛼  Introduction to thermodynamics Degree Industrial Technologies Engineering Water expansion In general, all of the materials undergo a thermal expansion on increasing the temperature. However, water exhibits a different behavior between 0 and 4 oC It undergoes a thermal expansion on decreasing For 0 < T < 4 oC temperature It undergoes a thermal compression on increasing temperature r(kg/l) Due to this feature of the water, only the water of the surface of the lakes freezes and the button of the lake remains unfrozen 1 T1 (Ice water) FA T1 < T2 T(oC) T2 (Liquid water) W 1 2 3 4 𝐹𝐴 = 𝑚𝑙𝑖𝑞𝑢𝑖𝑑 𝑔 = 𝜌2𝑉𝑔 The ice remains As 𝜌1 𝑖𝑐𝑒 < 𝜌2 𝑙𝑖𝑞𝑢𝑖𝑑 𝐹𝐴 > 𝑊 on the surface 𝑊 = 𝑚𝑖𝑐𝑒 𝑔 = 𝜌1𝑉𝑔 Note: FA is the force exerted by the water on the ice block due to the Archimedes principle Introduction to thermodynamics Degree Industrial Technologies Engineering Other thermal coefficients The thermal coefficient of compressibility (at constant temperature) is defined as : 1 𝜕𝑉 𝜕𝐿𝑛𝑉 𝐾𝑇 = − =− 𝑉 𝜕𝑃 𝑇 𝜕𝑃 𝑇 It is possible to relate the change of volume of a substance with the volume thermal coefficient of expansion (at constant pressure) and with the thermal coefficient of compressibility (at constant temperature) 𝜕𝑉 𝜕𝑉 As V=V(P,T) 𝑑𝑉 = 𝑑𝑃 + 𝑑𝑇 𝜕𝑃 𝑇 𝜕𝑇 𝑃 1 𝜕𝑉 1 𝜕𝑉 It is taken into account 𝐾𝑇 = − 𝛽= 𝑉 𝜕𝑃 𝑇 𝑉 𝜕𝑇 𝑃 𝑑𝑉 = −𝐾𝑇 𝑉𝑑𝑃 + 𝛽𝑉𝑑𝑇 Δ𝑉 = −𝐾𝑇 𝑉Δ𝑃 + 𝛽𝑉Δ𝑇 The piezothermal coefficient : It establishes the change of the pressure with the temperature on keeping the volume constant 1 𝜕𝑃 𝜕𝐿𝑛𝑃 𝛾= = 𝑃 𝜕𝑇 𝑉 𝜕𝑇 𝑉